Exponential stability of nonlinear positive systems on time scales

Bartosiewicz, Zbigniew

doi:10.1016/j.nahs.2019.02.011

Cited by 24 publications

(11 citation statements)

References 14 publications

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“…However, little attention has been paid to stability problem of positive systems on time scales. To the best of our knowledge, there are only two literatures focused on this problem 38,39 . However, the considered systems in 38,39 are trivial.…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, there are only two literatures focused on this problem 38,39 . However, the considered systems in 38,39 are trivial. In, 38 linear time‐invariant positive systems on time scales were investigated.…”

Section: Introductionmentioning

confidence: 99%

“…In, 38 linear time‐invariant positive systems on time scales were investigated. And a general class of nonlinear positive systems on time scales were addressed in 39 . Therefore, lots of stability problems about positive systems on time scales still remain to be solved.…”

Section: Introductionmentioning

confidence: 99%

“…(i) This paper first investigates the positive switched impulsive system with delay on time scales, which is more challenging than those in the literatures 38,39 . Moreover, the stability problem of the considered system is addressed from two cases: the switching behavior and the impulsive phenomenon happen synchronously and asynchronously.…”

Section: Introductionmentioning

confidence: 99%

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Stability analysis of positive switched impulsive systems with delay on time scales

Lü

Wang

et al. 2020

Intl J Robust & Nonlinear

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In this paper, stability analysis of positive switched impulsive systems with delay on time scales is investigated. Firstly, a necessary and sufficient condition is provided for positivity of the considered systems. Secondly, by respectively considering the situations that the switching behavior and the impulsive phenomenon happen simultaneously and asynchronously, the uniformly asymptotical stability problems are addressed. Then, based on the linear copositive Lyapunov-Krasovskii functional technique, two less conservative stability criteria are proposed. It are shown that these two criteria allow the timescale derivative of related Lyapunov functionals to be non-negative at some intervals. Moreover, these two criteria are effective for not only continuous cases and discrete cases, but also some other positive switched impulsive systems on general uniform or hybrid time domains. Finally, a non-continuous and non-discrete example is given to illustrate the effectiveness of the theoretic results. K E Y W O R D S positive switched impulsive systems, stability, time scale 1 INTRODUCTION Positive systems play an important role in many practical areas, such as economics, populations dynamics, ecology and Boolean network, etc. 1-7 The distinctive characteristics of this kind of systems are that their state variables, inputs and outputs take only nonnegative values. The common approaches for stability analysis of linear time-invariant positive systems are the Hurwitz criterion, the Routh criterion, and the Lyapunov technique (including the linear copositive Lyapunov function theorem and the diagonal quadratic Lyapunov function method). 2 And the Lyapunov function approach has been successfully introduced to stability analysis of nonlinear or delayed positive systems. 8,9 Recently, several other kinds of positive systems have also received continued attention, such as switched positive systems, 10-14 impulsive positive systems, 15-19 and switched impulsive positive systems, 20-23 etc. When studying switched impulsive positive systems, a natural question is that do the switching behavior and the impulsive phenomenon happen simultaneously or asynchronously? Unfortunately, the literatures 21-23 only consider the former. Actually, from the viewpoint of reality, it is more reasonable to consider the latter. Moreover, in, 21-23 the necessary conditions for analyzing stability of switched impulsive positive systems are to construct Lyapunov functionals with time-derivative or time-shift being strictly negative. However, this requirement can be surely weakened by the ideas in. 24,25 Generally speaking, continuous positive systems and discrete positive systems are always investigated separately. As far back as 1937, the work 26 stated that a major task of mathematics was to harmonize the continuous and the discrete, to

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Section: Introductionmentioning

confidence: 99%